This paper gives a detailed description of the design and structure
of the integral knot concordance group and solves the realization prob-
lem for the algebraic invariants described by J. Levine in his fundamen-
tal paper, "Invariants of Knot Cobordism", [L2] in which he demonstrates
their sufficiency. The geometric problem underlying and motivating the
algebraic discussion of this paper is the classification of spherical
knots under the equivalence relation of concordance: A smooth, oriented
manifold pair (M ,K ) , where K is a codimension two embedded sub-
manifold of M , is a spherical knot if M and K are homotopy spheres.
Two such pairs with the same ambient manifold M are concordant if
there is a proper oriented codimension two h-cobordism H in M x
[0,1] (with dH c d(MXI) such that the restriction to M X {0,1} is
the pairs (M,Kn) and (M,-K-, )) (where -denotes the opposite orienta-
tion). This is the geometric knot concordance group C- (s) first
introduced by Fox and Milnor [FM] for the case n = 1 . Kervaire com-
puted that the group was trivial for n even in [Kl] and Levine defined
an algebraic obstruction group (which we denote C (^) after Kervaire
[K2]) which he demonstrated was isomorphic to the geometric group
^2 +1^ for e = (""^ provided n 1 in [Ll]. (For n = 1 there
is only a surjection from the geometric group.) In [L2], Levine computed
the analogous group, C (Q) over the field of rational numbers (with
the aid of a computation of Milnor [Ml]) and observed that the integral
group injected into the rational group.
The group arises in a concrete geometric manner from the classical
Seifert pairing. If (M,K) is a spherical knot, then K is the
boundary of a framed codimension one submanifold V of M . The inter-
section pairing on the middle dimensional homology of V is e =
symmetric if n = 2k - 1 and is unimodular (has a unit determinant)
Received by the editor November 1, 1976.
This research was partially supported by an NSF Grant.
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